# Functional Analysis by M Usman Hamid and Zeeshan Ahmad

These notes are send by Muhammad Usman Hamid and written by Muhammad Usman Hamid and Zeeshan Ahmad. We are really very thankful to him for providing these notes and appreciate his effort to publish these notes on MathCity.org. Usman is dedicated and committed mathematician, who is working very hard for better understanding of mathematics to it readers.

Functional analysis is an abstract branch of mathematics that originated from classical analysis. It deals with analysis of functional (functions of functions). It concerned with infinite dimensional vector spaces (mainly function space) and mappings between them. It deals with abstract spaces and different operators define on these spaces. Its development started about eighty years ago, and nowadays functional analytic methods and results are important in various fields of mathematics and its applications. The impetus came from linear algebra, linear ordinary and partial differential equations, calculus of variations, approximation theory and, in particular, linear integral equations, whose theory had the greatest effect on the development and promotion of the modern ideas.

 Name Functional Analysis M Usman Hamid and Zeeshan Ahmad 278 pages PDF (see Software section for PDF Reader) 8.59 mB

In this section, the main heading are given below.

Metric Spaces: A quick review, completeness and convergence, completion,

Normed Spaces: linear spaces, normed spaces, difference between a metric and a normed space, Banach spaces, bounded and continuous linear operators and functionals, dual spaces, finite dimensional spaces, F. Riesz Lemma, the Hahn-Banach theorem, the HB theorem for complex spaces, The HB theorem for normed spaces, the open mapping theorem, the closed graph theorem, uniform boundedness principle and its applications.

Inner-Product Spaces: Inner-Product space, Hilbert Space, orthogonal and orthonormal sets, orthogonal complements, Gram-Schmidt orthogonalization process, representation of functionals, Reiz-representation theorem, and weak convergence.

Banach-Fixed-Point Theorem: Applications in differential and integral equations.